Power requirements for large-amplitude flapping flight

Journal Article

In this paper, a method is presented for computing the circulation distribution along the span of a flapping wing that minimizes the power required to generate a prescribed lift and thrust. The power is composed of three parts: useful thrust power, induced power, and profile power. Here, the thrust and induced power are expressed in terms of the Kelvin impulse and kinetic energy associated with the sheet of trailing and shed vorticity left behind the flapping wing. The profile power is computed using a quasisteady approximation of the two-dimensional viscous drag polar at each spanwise station of the wing. A variational principle is then formed to determine the necessary conditions for the circulation distribution to be optimal. Included in the variational principle is a constraint that the wing not stall. This Variational principle, which is essentially the viscous extension of the well-known Betz criterion for optimal propellers, is discretized using a vortex-lattice model of the wake, and the optimum solution is computed numerically. The present method is used to analyze a conventional propeller as well as a rigid wing in forward-flight flapping about a hinge point on the longitudinal axis.